CN102054122A - Haptic texture rendering method based on practical measurement - Google Patents

Abstract

The invention discloses a force-tactile expression method based on the actually measured texture. The resultant force is output to the operator as a contact force. The normal texture force is obtained by measuring the real pressure across the textured surface. A mechanical arm with a constant force and a pressure sensor at the bottom is used to swipe across the surface of the textured material at a constant speed, and the data is collected at the same time. The data is corrected for error items, smoothed, and voltage pressure The normal force is converted into the normal texture force after subtracting the constant force value of the manipulator; the normal binding force is modeled as a spring damping model; the tangential friction force modeling combines the two stages of static friction and sliding friction, and the static friction stage is The modulus is the product of the maximum static friction force and the sine function, and the dynamic friction coefficient in the dynamic friction stage is calculated from the normal texture force reflecting the degree of texture unevenness, which improves the realism of texture expression.

Description

Power haptic rendering method based on the texture of actual measurement

Technical field

The present invention relates to the power haptic rendering method of texture, specifically proposed a kind of expression of the texture power based on true measurement.

Background technology

Along with the continuous development of virtual reality technology, power haptic rendering technology is remarkable day by day for the importance of pseudo operation.Texture is the comprehensive description of characteristic such as the microcosmos geometric shape, roughness of body surface, and the power haptic rendering research of texture is very important.The power haptic rendering of texture is that the texture information of body surface is fed back to the operator by power tactile sense reproduction equipment, can make operator's physical attribute of perception texture truly, thereby make the operator obtain the strong feeling of immersion and the sense of reality, this is the final goal that texture force touch is expressed.

At present, mainly according to the height profile feature of grain surface, the contact force computation model when foundation contacts with grain surface reproduces texture power to the operator by power tactile sense reproduction equipment the power haptic rendering of texture, thereby reproduces the concave-convex sense of texture.The method for building up of the power haptic model of texture has three kinds usually: (1) extracts grain surface feature contour feature from image, sets up texture force touch and expresses model; (2), set up texture force touch and express model based on geometrical constraint and physical model; (3), produce fictitious force sense of touch texture power mapping model by the concavo-convex profile of actual measurement grain surface.

Power sense of touch texture based on Flame Image Process is expressed model, and this is the focus in present reproducing texture force touch field.This model need extract the feature of grain surface from image, not only image processing algorithm is had relatively high expectations, and is also very high to the accuracy requirement of image itself.But usually because the quality of image itself can not guarantee that the grain surface height that comes out by image calculation can not reflect texture information truly.

The texture force touch set up based on geometrical constraint is expressed the interaction that model has been realized power between two polygon models.Express model based on the power sense of touch texture of physical model and mainly comprise sinusoidal model, probabilistic model etc.Sinusoidal model is modeled as the sinusoidal signal that the roughness with article surface vein changes with the texture contact force, and body surface is coarse more, and the amplitude of cyclical signal is just big more, and texture is fine and closely woven more, and then the frequency of periodic signal is high more.This method can reflect the characteristic of texture to a certain extent preferably, but the texture kind that deterministic models can be expressed is limited, only can make force feeling that tangible discrimination is arranged, but lacks the sense of reality, can not texture tactility that is virtually reality like reality.Research on stochastic model mainly concentrates at random the descriptive statistics, and statistical method comprises auto-correlation, marginal density, feature histogram and random field etc.The contact force that different textures produces has different numerical characteristics, and by changing the value of numerical characteristic, the roughness of superficial makings can be changed.But only can express the roughness of texture based on the power haptic rendering of probabilistic model, can not reproduce the real concavo-convex and tendency that grain surface exists.

The sensor or the instrument that are used for the surface measurements texture mainly are divided into two kinds: a kind of is contactless, without any contact, obtains surface topography information by optical technology between object and the surveying instrument.As the light probe interferometric method, adopt very thin focused light probe incident measured surface, the height change of body surface causes the variation of reference light and measuring light optical path difference, by the phase bit comparison, can obtain the profile on surface.But measuring method is had relatively high expectations to the cleanliness of measured surface, and dust is bigger to its measurement result influence.Another kind is a contact, and probe contacts with body surface, measures the outline data of grain surface, again the geometric profile data is converted to texture power.The most widely used contact type measurement instrument is the contact pin type contourgraph, and they generally adopt adamas probe scanning measured surface.People such as Vasudevan propose with the probe of PHANToM hand controller the grain surface profile to be explored, and measure the displacement of vertical direction probe.By profile information being converted to after the texture power, using various power haptic models and texture is expressed carry out the power sense of touch and play up.The method that also has is the texture power when measuring staff and initiatively slipping over grain surface, at this moment needs to reject the staff subjectivity and applies the influence of power to measurement result.

Summary of the invention

The invention provides and a kind ofly can improve the power haptic rendering method that texture power is expressed authenticity based on the texture of actual measurement.

A kind of power haptic rendering method of the texture based on actual measurement is characterized in that:

The lever convergence virtual texture surface of steering force tactile sense reproduction equipment, establish with the power output of grain surface before bumping be 0, after bumping, produce normal direction texture power with grain surface

The normal direction binding force

And tangent direction friction power

At last, with normal direction texture power

The normal direction binding force

And tangent direction friction power

Make a concerted effort as contact force

And with this contact force

As power output;

Described normal direction texture power

Generate by following method:

Step 1 is chosen real texture materials;

Step 2 is installed in pressure transducer the bottom of mechanical arm;

It is at the uniform velocity mobile on the texture materials surface that step 3 has the mechanical arm of pressure transducer, and mechanical arm applies constant force F _In, pressure transducer being contacted with the texture materials surface, and obtain the data of sampling with sampling apparatus, the sampling rate of data is 1000Hz, i.e. time resolution ax/t ₀Be 0.001s, record mechanical arm tangential movement speed v _sWith displacement S, sampling number

Step 4 is carried out data processing with the magnitude of voltage of measured pressure transducer, comprising:

(1) data pre-service:

If the arbitrary voltage sample value in the N that the is surveyed sampled point is less than voltage reference value U ₀, then with voltage reference value U ₀Replace described less than voltage reference value U ₀Arbitrary voltage sample value, at this moment, the voltage sample value of N sampled point is u ₀₁, u ₀₂... u _0N, adopt 5 triple smoothings that the voltage sample value is carried out smoothing processing, concrete formula is:

$\left\{\begin{array}{c}{u}_1=\frac{1}{70}[69u_{01}+4(u_{02}+u_{04})-6u_{03}-u_{05}]\\ u_2=\frac{1}{35}[2(u_{01}+u_{05})+27u_{02}+12u_{03}-8u_{04}]\\ u_i=\frac{1}{35}[-3(u_{0i-2}+u_{0i+2})+12(u_{0i-1}+u_{0i+1})+17u_{0i}]\\ u_{N-1}=\frac{1}{35}[2(u_{0N-4}+u_{0N})-8u_{0N-3}+12u_{0N-2}+27u_{0N-1}]\\ u_N=\frac{1}{70}[-u_{0N-4}+4(u_{0N-3}+u_{0N-1})-6u_{0N-2}+69u_{0N}]\end{array}\right.$

In the formula, u ₁, u ₂... u _NBe the voltage sample value after level and smooth, i=3,4 ... N-2;

(2) with the voltage sample value u after level and smooth ₁, u ₂... u _NBe converted to force value F ₁, F ₂... F _N, according to the circuit theory conversion formula be J=1,2 ... N, wherein, U ₀Be voltage reference value, L is a transducer sensitivity, and G is the constant by the circuit decision;

(3) with force value F _jDeduct the constant force value F that mechanical arm applies _In, obtain reflecting the concavo-convex different force value of grain surface, i.e. normal direction texture power

Normal direction texture power

Sequence table be shown

J=1,2 ... N, formula is:

$f_{\mathrm{texture}}^{\→}\left(j\right)=F_j-F_{\mathrm{in}},j=\mathrm{1,2},...N$

Step 5 is with texture power

In virtual scene coordinate system Oxyz, express, be expressed as

(1) initialization power tactile sense reproduction equipment and virtual scene; Virtual scene coordinate system Oxyz is set, supposes that the coordinate of the point of impingement under virtual scene space coordinates Oxyz is (x, y, z), the zone in the coordinate system { 0≤x≤X, 0≤y≤Y, z=Z} is the texture materials surface, zone { 0≤x≤X, 0≤y≤Y, z＞Z} is the texture materials outside, zone { 0≤x≤X, 0≤y≤Y, z＜Z} is texture materials inside, and X, Y represent the boundary position coordinate of texture region at x axle, y axle respectively, and X＞0, Y＞0, Z represents the position coordinates of texture plane on the z axle;

(2) with N-1 plane

With area of space z≤Z} average mark is slit into N zonule for 0≤x≤X, 0≤y≤Y, in arbitrary zonule, texture power

All be a fixed value, along the z direction of principal axis, concrete formula is:

$f_{\mathrm{texture}}^{\&RightArrow;}(x,y,z)=f_{\mathrm{texture}}^{\&RightArrow;}\left(j\right),$ When $(x,y,z)\&Element;\{\frac{(j-1)X}{N}<x\&le;\frac{\mathrm{jX}}{N},0\&le;y\&le;Y,z\&le;Z|j=\mathrm{1,2},...N\}$

Described normal direction binding force

Generate by following method:

The normal direction binding force

Be modeled as the spring damping model

Along the z direction of principal axis, concrete formula is:

$f_{\mathrm{constraint}}^{\&RightArrow;}(x,y,z)=k*|z-Z|+b*(-\frac{\mathrm{dz}}{\mathrm{dt}}),z\&le;Z;$

K*|z-Z| is the spring deformation item, k is a stiffness coefficient, expression texture materials surface collided and the extruding situation under deformation extent, relevant with the attribute on texture materials surface, the k span is 0-1 (N/mm), and k is 1 (N/mm) when object is perfect rigidity, and k was 0 when object was full flexible, Z is the position coordinates of texture plane on the z axle, | z-Z| is the paracentesis depth of Z-direction;

Be damping term,

For virtual probe along the axial instantaneous velocity of z, concrete computing formula is

Δ t is the renewal rate of power tactile sense reproduction equipment, the viscous damping coefficient of b for being directly proportional with the energy consumption of texture materials surface deformation, the b span is 0.1-0.5, b is big more, the difficult more generation deformation in texture materials surface, the b value is 0.5 when object is perfect rigidity, and the b value was 0.1 when object was full flexible;

Described tangent direction friction power

Generate by following method:

Tangent direction friction power

Direction opposite with virtual probe in the relative motion or the relative motion trend direction of X-Y plane, the computing formula of friction force under the virtual scene coordinate system

For:

$f_{\mathrm{friction}}^{\&RightArrow;}(x,y,z)=\left\{\begin{array}{c}\mathrm{\&mu;}(x,y,z)F_N(x,y,z),v\&le;-\left|v_r\right|\\ f_{\mathrm{ms}}\sin (-\frac{\mathrm{\&pi;}}{2\left|v_r\right|}v)={\mathrm{\&mu;}}_0(x,y,z)F_N(x,y,z)\sin (-\frac{\mathrm{\&pi;}}{2\left|v_r\right|}v),v<\left|v_r\right|\\ -\mathrm{\&mu;}(x,y,z)F_N(x,y,z),v\&GreaterEqual;\left|v_r\right|\end{array}\right.$

$=\left\{\begin{array}{c}\mathrm{\&mu;}(x,y,z)(f_{\mathrm{texture}}^{\&RightArrow;}(x,y,z)+f_{\mathrm{constraint}}^{\&RightArrow;}(x,y,z)),v\&le;-\left|v_r\right|\\ {\mathrm{\&mu;}}_0(x,y,z)(f_{\mathrm{texture}}^{\&RightArrow;}(x,y,z)+f_{\mathrm{constraint}}^{\&RightArrow;}(x,y,z))\sin (-\frac{\mathrm{\&pi;}}{2\left|v_r\right|}v),v<\left|v_r\right|\\ -\mathrm{\&mu;}(x,y,z)(f_{\mathrm{texture}}^{\&RightArrow;}(x,y,z)+f_{\mathrm{constraint}}^{\&RightArrow;}(x,y,z)),v\&GreaterEqual;\left|v_r\right|\end{array}\right.$

V is the translational speed of virtual probe at X-Y plane in the formula, and computing formula is

v _rBe the speed limit of virtual probe at X-Y plane, when v＜| v _r| the time, being approximate relative static conditions, friction force is modeled as stiction

f _Ms=μ ₀(x, y, z) F _N(x, y z) are maximum static friction force, μ ₀(x, y z) are the coefficient of static friction of the point of impingement, F _N(x, y, z) expression normal pressure, i.e. normal direction texture power With the normal direction binding force

Vector and; When movement speed v 〉=| v _r| or v≤-| v _r|, friction force is modeled as force of sliding friction, F _N(z) the expression normal pressure is normal direction texture power for x, y

With the normal direction binding force

Vector and, μ (x, y z) are dynamic friction factor, and be relevant with the degree of roughness of the material of surface of contact and surface of contact, μ (expression formula z) is for x, y:

μ (x, y, z)=ρ r (z), wherein ρ is the coefficient of roughness of texture materials for x, y, r (x, y z) are the local roughness coefficient of the point of impingement, and be relevant with the local concavo-convex degree of this point of impingement, r (calculating z) is obtained by following formula for x, y:

C wherein _nBe flexible strategy, value and distance

Be inversely proportional to, and

The value of q is generally 2 or 3; Coefficient of static friction μ in the model ₀(x, y, z) more than or equal to kinetic friction coefficient μ (x, y, z).

Advantage of the present invention

1, the pressure of perception texture is measured the surface characteristics that can embody texture, concavo-convex as texture, with the data process data processing of gathering, promptly correct mistakes, level and smooth, voltage force value conversion, be converted to virtual texture power after deducting mechanical arm constant force value, because virtual texture power changes consistent characteristics with true grain surface contact force, therefore the texture power that will from true measurement environment, extract be used in the modeling of power and firmly tactile sense reproduction equipment to export be the reliable means that realizes that true texture is expressed;

2, the measuring method science of pressure is simple, the mechanical arm that the utilization end has a pressure transducer evenly grain surface slide and the mechanical arm application of force constant, both got rid of of the influence of staff application of force inequality, reached the correspondence in sampling time and space again, and be convenient to analyze for measured pressure value;

3, contact force model is modeled as the vector addition model of normal direction texture power, normal direction binding force, tangent direction friction power, this meets the general characteristic that staff touches the perception texture, normal direction texture power has embodied the rough tendency of texture, the normal direction binding force has embodied the material properties of object, tangent direction friction power has embodied the degree of roughness of article surface vein, and these all are the requisite key elements that realizes the texture truly expressed;

4, the binding force model is the spring damping model, embodied article surface vein collided and the extruding situation under deformation extent, and damping term is directly proportional with the movement velocity of normal direction, reflected of the influence of the movement velocity of normal direction when the actual touch grain surface to the size variation of contact force, normal velocity is big more, the contact force of being experienced is big more, and the bump impression is obvious more;

5, for the model of the power that makes foundation more during real simulated perception texture the trend of power change, the combined action of stiction and force of sliding friction has been considered in the modeling of friction force, when static relatively between virtual probe and the grain surface but when relative motion trend is arranged, friction force is modeled as nonlinear stiction, it is the product of maximum static friction force and sine function, increase gradually when movement velocity but also do not reach speed in limited time, stiction increases with the increase of speed, after speed reached the speed limit, friction force just was converted into force of sliding friction;

6, the kinetic friction coefficient in the force of sliding friction is not only relevant with the material coefficient of roughness on texture materials surface, also relevant with the local roughness coefficient, the local roughness coefficient is by obtaining apart from square process weighted sum of the point of impingement apart from the difference of measurement point that does not wait and point of impingement texture power around the point of impingement, the degree of fluctuation that has reflected texture, meet the information that staff slips over the power of being fed back of grain surface, improved the sense of reality that texture is expressed.

Description of drawings

Fig. 1 is a system chart of the present invention;

Fig. 2 is a normal direction texture power of the present invention Product process figure;

Fig. 3 is an experiment synoptic diagram of the present invention;

Fig. 4 is a contact force of the present invention

Model;

Fig. 5 is a virtual scene coordinate system synoptic diagram of the present invention;

Fig. 6 is the collision response synoptic diagram under the virtual scene coordinate of the present invention;

Specific embodiments

Be described in further detail below in conjunction with the enforcement of accompanying drawing technical scheme:

With reference to Fig. 1, a kind of power haptic rendering method of the texture based on actual measurement is characterized in that:

The lever convergence virtual texture surface of steering force tactile sense reproduction equipment, establish with the power output of grain surface before bumping be 0, after bumping, produce normal direction texture power with grain surface

The normal direction binding force

And tangent direction friction power

At last, with normal direction texture power

The normal direction binding force And tangent direction friction power

Make a concerted effort as contact force

And with this contact force Be power output, with reference to Fig. 4, promptly

$f_{\mathrm{contact}}^{\&RightArrow;}=f_{\mathrm{texture}}^{\&RightArrow;}+f_{\mathrm{constraint}}^{\&RightArrow;}+f_{\mathrm{friction}}^{\&RightArrow;};$

With reference to Fig. 2, described normal direction texture power

Generate by following method:

Step 1 is chosen real texture materials: Art Stone;

Step 2 is installed in pressure transducer the bottom of mechanical arm;

Step 3 makes the mechanical arm that has pressure transducer at the uniform velocity mobile at grain surface, and as shown in Figure 3: mechanical arm applies constant force F _In=1N makes pressure transducer contact with the texture materials surface, and obtains the data of sampling with sampling apparatus, and the sampling rate of data is 1000Hz, i.e. time resolution ax/t ₀Be 0.001s, record mechanical arm tangential movement speed v _s=10mm/s and displacement S=100mm, sampling number

Step 4 is carried out data processing with the magnitude of voltage of measured pressure transducer, comprising:

(1) data pre-service:

If the arbitrary voltage sample value in the N=10001 that the is surveyed sampled point is less than voltage reference value U ₀, then with voltage reference value U ₀Replace described less than voltage reference value U ₀Arbitrary voltage sample value, U in this circuit ₀=2.42V, at this moment, the voltage sample value of N sampled point is u ₀₁, u ₀₂... u _0N, adopt 5 triple smoothings that the voltage sample value is carried out smoothing processing, concrete formula is:

$\left\{\begin{array}{c}{u}_1=\frac{1}{70}[69u_{01}+4(u_{02}+u_{04})-6u_{03}-u_{05}]\\ u_2=\frac{1}{35}[2(u_{01}+u_{05})+27u_{02}+12u_{03}-8u_{04}]\\ u_i=\frac{1}{35}[-3(u_{0i-2}+u_{0i+2})+12(u_{0i-1}+u_{0i+1})+17u_{0i}]\\ u_{N-1}=\frac{1}{35}[2(u_{0N-4}+u_{0N})-8u_{0N-3}+12u_{0N-2}+27u_{0N-1}]\\ u_N=\frac{1}{70}[-u_{0N-4}+4(u_{0N-3}+u_{0N-1})-6u_{0N-2}+69u_{0N}]\end{array}\right.$

In the formula, u ₁, u ₂... u _NBe the voltage sample value after level and smooth, i=3,4 ... N-2;

(2) with the voltage sample value u after level and smooth ₁, u ₂... u _NBe converted to force value F ₁, F ₂... F _N, according to the circuit theory conversion formula be

J=1,2 ... N, wherein, U ₀=2.42V, L=0.12mv/g, G=17.467;

(3) with force value F _jDeduct the constant force value F that mechanical arm applies _In, obtain reflecting the concavo-convex different force value of grain surface, i.e. normal direction texture power

Normal direction texture power

Sequence table be shown J=1,2 ... N, formula is:

$f_{\mathrm{texture}}^{\&RightArrow;}\left(j\right)=F_j-F_{\mathrm{in}},j=\mathrm{1,2},...N$

If texture power

Less than 0, then show the contact point depression, if texture power

Greater than 0, expression contact point projection;

Step 5 is with texture power

In virtual scene coordinate system 0xyz, express, be expressed as

(1) initialization power tactile sense reproduction equipment and virtual scene; With reference to Fig. 5, virtual scene coordinate system 0xyz is set, suppose that the coordinate of the point of impingement under virtual scene space coordinates 0xyz is (x, y, z), { 0≤x≤X of the zone in the coordinate system, 0≤y≤Y, z=Z} are the texture materials surface, zone { 0≤x≤X, 0≤y≤Y, z＞Z} are the texture materials outside, zone { 0≤x≤X, 0≤y≤Y, z＜Z} are texture materials inside, wherein get X=200, Y=200, Z=0;

(2) with N-1 plane

With area of space 0≤x≤X, 0≤y≤Y, z≤Z} are divided into N zonule, in arbitrary zonule, texture power

All be a fixed value, along the z direction of principal axis, concrete formula is:

$f_{\mathrm{texture}}^{\&RightArrow;}(x,y,z)=f_{\mathrm{texture}}^{\&RightArrow;}\left(j\right),$ When $(x,y,z)\&Element;\{\frac{(j-1)X}{N}<x\&le;\frac{\mathrm{jX}}{N},0\&le;y\&le;Y,z\&le;Z|j=\mathrm{1,2},...N\}$

With reference to Fig. 6, described normal direction binding force

Generate by following method:

Binding force

Be modeled as the spring damping model

Along the z direction of principal axis:

$f_{\mathrm{constraint}}^{\&RightArrow;}(x,y,z)=k*|z-Z|+b*(-\frac{\mathrm{dz}}{\mathrm{dt}}),z\&le;Z;$

K*|z-Z| is the spring deformation item, and k is a stiffness coefficient, expression texture materials surface collided and the extruding situation under deformation extent, relevant with the attribute on texture materials surface, k gets 0.8, and Z is the position coordinates of texture plane on the z axle, | z-Z| is the paracentesis depth of Z-direction;

Be damping term,

For virtual probe along the axial instantaneous velocity of z, concrete computing formula is

Δ t is the renewal rate of power tactile sense reproduction equipment, the viscous damping coefficient of b for being directly proportional with the energy consumption of texture materials surface deformation, and the b span is 0.1-0.5, b is big more, the difficult more generation deformation in texture materials surface, b gets 0.4;

Described tangent direction friction power

Generate by following method:

Tangent direction friction power

Direction opposite with virtual probe in the relative motion or the relative motion trend direction of X-Y plane, the computing formula of friction force under the virtual scene coordinate system

For:

$f_{\mathrm{friction}}^{\&RightArrow;}(x,y,z)=\left\{\begin{array}{c}\mathrm{\&mu;}(x,y,z)F_N(x,y,z),v\&le;-\left|v_r\right|\\ f_{\mathrm{ms}}\sin (-\frac{\mathrm{\&pi;}}{2\left|v_r\right|}v)={\mathrm{\&mu;}}_0(x,y,z)F_N(x,y,z)\sin (-\frac{\mathrm{\&pi;}}{2\left|v_r\right|}v),v<\left|v_r\right|\\ -\mathrm{\&mu;}(x,y,z)F_N(x,y,z),v\&GreaterEqual;\left|v_r\right|\end{array}\right.$

$=\left\{\begin{array}{c}\mathrm{\&mu;}(x,y,z)(f_{\mathrm{texture}}^{\&RightArrow;}(x,y,z)+f_{\mathrm{constraint}}^{\&RightArrow;}(x,y,z)),v\&le;-\left|v_r\right|\\ {\mathrm{\&mu;}}_0(x,y,z)(f_{\mathrm{texture}}^{\&RightArrow;}(x,y,z)+f_{\mathrm{constraint}}^{\&RightArrow;}(x,y,z))\sin (-\frac{\mathrm{\&pi;}}{2\left|v_r\right|}v),v<\left|v_r\right|\\ -\mathrm{\&mu;}(x,y,z)(f_{\mathrm{texture}}^{\&RightArrow;}(x,y,z)+f_{\mathrm{constraint}}^{\&RightArrow;}(x,y,z)),v\&GreaterEqual;\left|v_r\right|\end{array}\right.$

V is the translational speed of virtual probe at X-Y plane, and computing formula is v _rRule of thumb be worth under the virtual scene coordinate system and get 0.072/s, be equivalent to 8mm/s, when v＜| v _r| the time, being approximate relative static conditions, friction force is modeled as stiction

f _Ms=μ ₀(x, y, z) F _N(x, y z) are maximum static friction force, μ ₀(x, y z) are the coefficient of static friction of the point of impingement, F _N(x, y, z) expression normal pressure, i.e. normal direction texture power

With the normal direction binding force

Vector and; When movement speed v 〉=| v _r| or v≤-| v _r|, friction force is modeled as force of sliding friction, F _N(x, y, z) expression normal pressure, μ (x, y z) are dynamic friction factor, and be relevant with the degree of roughness of the material of surface of contact and surface of contact, μ (expression formula z) is for x, y:

μ (x, y, z)=ρ r (z), wherein ρ is the coefficient of roughness of texture materials for x, y, ρ=0.4, r (x, y z) are the local roughness coefficient of contact point, and be relevant with the local concavo-convex degree of this contact point, r (calculating z) is obtained by following formula for x, y:

$r(x,y,z)=\underset{n\&NotEqual;0}{\overset{q}{\underset{n=-q}{\mathrm{\&Sigma;}}}}{c}_n{|F_{\mathrm{texture}}^{\&RightArrow;}(x-\frac{\mathrm{nX}}{N},y,z)-F_{\mathrm{texture}}^{\&RightArrow;}(x,y,z)|}^2$

Work as q=2,

Get coefficient of static friction μ in the model ₀(x, y, z) equal kinetic friction coefficient μ (x, y, z);

The work space of the lever of the moving area of virtual probe and power tactile sense reproduction equipment is set up the spatial mappings relation in the virtual scene coordinate system, wherein, lever positional information (the Px of power tactile sense reproduction equipment output, Py, Pz) obtain by the canonical function dhdGetPosition () among the power tactile sense reproduction equipment standard function library API, (x in the virtual scene coordinate system, y, z) by the lever coordinate figure (Px of power tactile sense reproduction equipment, Py, Pz) get through coordinate convergent-divergent and translation, concrete ratio variation relation is as follows:

x＝9*Px，y＝9*Py，z＝5*Pz-5

At last, normal direction texture power in the virtual scene The normal direction binding force

And tangent direction friction power

Make a concerted effort by the output of the canonical function dhdSetForce () among the power tactile sense reproduction equipment standard function library API.

Claims (1)

1. A force-tactile expression method based on an actual measured texture, characterized in that: Manipulation force The operating handle of the tactile reproduction device approaches the virtual texture surface, and the output force before colliding with the texture surface is set to 0. After colliding with the texture surface, the normal texture force is generated normal binding force

and tangential friction

Finally, with the normal texture force

normal binding force

and tangential friction

The resultant force of the contact force

and with this contact force

as an output force; The normal texture force

Generated by: Step 1 select the real texture material; Step 2 Install the pressure sensor on the bottom of the robotic arm; Step 3. The mechanical arm with the pressure sensor moves at a constant speed on the surface of the textured material, and the mechanical arm applies a constant force F _in to make the pressure sensor contact with the surface of the textured material, and use the sampling device to obtain the sampled data. The sampling rate of the data is 1000Hz, that is, the time The resolution Δt ₀ is 0.001s, record the horizontal motion velocity v _s and displacement S of the manipulator, and the number of sampling points Step 4 performs data processing on the measured voltage value of the pressure sensor, including: (1) Data preprocessing: If any voltage sampling value in the N sampling points measured is less than the voltage reference value U ₀ , then replace any voltage sampling value smaller than the voltage reference value U ₀ with the voltage reference value U ₀ , at this time, N The voltage sampling values of the sampling points are u ₀₁ , u ₀₂ , ... u _0N , and the voltage sampling values are smoothed by the five-point three-point smoothing method. The specific formula is: $\left\{\begin{array}{c}{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 70</span>}}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 69</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 01</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 02</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 04</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 6</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 03</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 05</span>}}<span\; class="notranslate">\; ]</span>\\ {\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 35</span>}}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 01</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 05</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 27</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 02</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 12</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 03</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 04</span>}}<span\; class="notranslate">\; ]</span>\\ {\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 35</span>}}<span\; class="notranslate">\; [</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 3</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 12</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 17</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; ]</span>\\ {\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 35</span>}}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 8</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 12</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 27</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ]</span>\\ {\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 70</span>}}<span\; class="notranslate">\; [</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 6</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 69</span>}{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; ]</span>\end{array}\right.$ In the formula, u ₁ , u ₂ ,...u _N are the smoothed voltage sampling values, i=3, 4,...N-2; (2) Convert the smoothed voltage sampling values u ₁ , u ₂ , ... u _N into pressure values F ₁ , F ₂ , ... F _N , according to the circuit principle, the conversion formula is

j=1, 2,...N, wherein, U ₀ is the voltage reference value, L is the sensor sensitivity, and G is a constant determined by the circuit; (3) Subtract the constant force value F _in applied by the mechanical arm from the pressure value F _j to obtain the pressure value reflecting the different unevenness of the textured surface, that is, the normal texture force

normal texture force

The sequence is expressed as j=1, 2,...N, the formula is: ${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; texture</span>}}^{<span\; class="notranslate">\; \&Right\; Arrow;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; in</span>}}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1,2</span>}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span><span\; class="notranslate">\; .</span>\mathrm{<span\; class="notranslate">\; N</span>}$ Step 5 will texture force

Expressed in the virtual scene coordinate system Oxyz, expressed as (1) Initialize the haptic reproduction device and the virtual scene; set the virtual scene coordinate system Oxyz, assume that the coordinates of the collision point in the virtual scene space coordinate system Oxyz are (x, y, z), and the area in the coordinate system {0≤x ≤X, 0≤y≤Y, z=Z} is the surface of the texture material, and the area {0≤x≤X, 0≤y≤Y, z>Z} is the outside of the texture material, and the area {0≤x≤X, 0 ≤y≤Y, z<Z} is inside the texture material, X and Y represent the boundary position coordinates of the texture area on the x-axis and y-axis respectively, and X>0, Y>0, Z represents the texture plane on the z-axis Position coordinates; (2) Use N-1 planes

Divide the space area {0≤x≤X, 0≤y≤Y, z≤Z} into N small areas on average, in any small area, the texture force

Both are a fixed value, along the z-axis direction, the specific formula is: $f_{\mathrm{texture}}^{\&Right\; Arrow;}(x,\mathrm{the\; y},z)=f_{\mathrm{texture}}^{\&Right\; Arrow;}\left(j\right),$ when $(x,\mathrm{the\; y},z)\&Element;\{\frac{(j-1)x}{N}<x\&le;\frac{\mathrm{wxya}}{N},0\&le;\mathrm{the\; y}\&le;Y,z\&le;Z|j=\mathrm{1,2},...N\}$ The normal binding force

Generated by: normal binding force

Modeled as a spring-damper model

Along the z-axis direction, the specific formula is: ${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; constraint</span>}}^{<span\; class="notranslate">\; \&Right\; Arrow;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; *</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; Z</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; b</span>}<span\; class="notranslate">\; *</span><span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; dz</span>}}{\mathrm{<span\; class="notranslate">\; dt</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; Z</span>}<span\; class="notranslate">\; ;</span>$ k*|z-Z| is the spring deformation item, k is the stiffness coefficient, which indicates the degree of deformation of the surface of the texture material when it is collided and squeezed, and is related to the properties of the surface of the texture material. The value range of k is 0-1(N/ mm), when the object is completely rigid, k is 1 (N/mm), when the object is completely flexible, k is 0, Z is the position coordinate of the texture plane on the z-axis, |z-Z| is the puncture depth in the Z-axis direction;

is the damping term,

is the instantaneous velocity of the virtual probe along the z-axis direction, and the specific calculation formula is

Δt is the update rate of the haptic reproduction device, b is the viscous damping coefficient proportional to the energy consumption of the surface deformation of the texture material, the range of b is 0.1-0.5, the larger b is, the less likely the surface of the texture material is to deform, when The value of b is 0.5 when the object is completely rigid, and the value of b is 0.1 when the object is completely flexible; The tangential friction

Generated by: Tangential friction

The direction of the virtual probe is opposite to the relative motion or relative motion trend direction of the virtual probe in the XY plane, and the calculation formula of the friction force in the virtual scene coordinate system

for: ${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; friction</span>}}^{<span\; class="notranslate">\; \&Right\; Arrow;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; \&mu;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; \&le;</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; |</span>\\ {\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; ms</span>}}\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; |</span>}\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&mu;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; |</span>}\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; <</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; |</span>\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&mu;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; N</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; \&Greater\; Equal;</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; |</span>\end{array}\right.$ $<span\; class="notranslate">\; =</span>\left\{\begin{array}{c}\mathrm{<span\; class="notranslate">\; \&mu;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; texture</span>}}^{<span\; class="notranslate">\; \&Right\; Arrow;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; constraint</span>}}^{<span\; class="notranslate">\; \&Right\; Arrow;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; \&le;</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; |</span>\\ {\mathrm{<span\; class="notranslate">\; \&mu;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; texture</span>}}^{<span\; class="notranslate">\; \&Right\; Arrow;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; constraint</span>}}^{<span\; class="notranslate">\; \&Right\; Arrow;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; sin</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; |</span>}\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; <</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; |</span>\\ <span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&mu;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; texture</span>}}^{<span\; class="notranslate">\; \&Right\; Arrow;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; constraint</span>}}^{<span\; class="notranslate">\; \&Right\; Arrow;</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; x</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; the\; y</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; z</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; v</span>}<span\; class="notranslate">\; \&Greater\; Equal;</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; |</span>\end{array}\right.$ In the formula, v is the moving speed of the virtual probe in the XY plane, and the calculation formula is

v _r is the rate limit of the virtual probe on the XY plane, when v<|v _r |, it is an approximate relative static state, and the friction is modeled as static friction

f _ms = μ ₀ (x, y, z) F _N (x, y, z) is the maximum static friction force, μ ₀ (x, y, z) is the static friction coefficient of the collision point, F _N (x, y, z ) represents the positive pressure, that is, the normal texture force and normal binding The vector sum of ; when the moving speed v≥|v _r | or v≤-|v _r |, the friction force is modeled as sliding friction force, F _N (x, y, z) represents the normal pressure, and is the normal texture force

and normal binding

The vector sum of μ(x, y, z) is the kinetic friction coefficient, which is related to the material of the contact surface and the roughness of the contact surface. The expression of μ(x, y, z) is: μ(x, y, z)=ρ r(x, y, z), where ρ is the roughness coefficient of the texture material, r(x, y, z) is the local roughness coefficient of the collision point, and the collision point’s The degree of local unevenness is related, and the calculation of r(x, y, z) is obtained by the following formula:

Where c _n is the weight, value and distance

inversely proportional to

The value of q is usually 2 or 3; the static friction coefficient μ ₀ (x, y, z) in the model is greater than or equal to the dynamic friction coefficient μ (x, y, z).

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